25 lines
855 B
ReStructuredText
25 lines
855 B
ReStructuredText
Discrete Fourier Transform
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Discrete Fourier Transform (DFT) is used to perform the Fourier transform of a discrete signal.
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The DFT of a discrete complex/real signal :math:`\vec{x}={x_0,...,x_{N-1}}` is:
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.. math::
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X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-j2\pi\frac{k}{N}n}
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.. note::
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In this case, :math:`\frac{k}{N}` can be seen as the frequency of the wave used to determined
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the :math:`k^{th}` frequency :math:`X_k`.
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See `here <https://www.youtube.com/watch?v=mkGsMWi_j4Q&t=282s>`__ for more infos.
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The inverse DFT of a discrete complex/real signal :math:`\vec{x}={x_0,...,x_{N-1}}` is:
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.. math::
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x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{j2\pi\frac{k}{N}n}
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.. note::
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If the signal is *real*, all the complex terms of the inverse DFT are suppose to cancel out.
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In depth example
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=================
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